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We correct some statements and proofs of K. S. Kedlaya [Local and global structure of connections on nonarchimedean curves, Compos. Math. 151 (2015), 1096–1156]. To summarize, Proposition 1.1.2 is false as written, and we provide here a corrected statement and proof (and a corresponding modification of Remark 1.1.3); the proofs of Theorem 2.3.17 and Theorem 3.8.16, which rely on Proposition 1.1.2, are corrected accordingly; some missing details in the proofs of Theorem 3.4.20 and Theorem 3.4.22 are filled in; and a few much more minor corrections are recorded.

We compute the complete set of candidates for the zeta function of a K
$3$
surface over
$\mathbb{F}_{2}$
consistent with the Weil and Tate conjectures, as well as the complete set of zeta functions of smooth quartic surfaces over
$\mathbb{F}_{2}$
. These sets differ substantially, but we do identify natural subsets which coincide. This gives some numerical evidence towards a Honda–Tate theorem for transcendental zeta functions of K
$3$
surfaces; such a result would refine a recent theorem of Taelman, in which one must allow an uncontrolled base field extension.

Consider a vector bundle with connection on a
$p$
-adic analytic curve in the sense of Berkovich. We collect some improvements and refinements of recent results on the structure of such connections, and on the convergence of local horizontal sections. This builds on work from the author’s 2010 book and on subsequent improvements by Baldassarri and by Poineau and Pulita. One key result exclusive to this paper is that the convergence polygon of a connection is locally constant around every type 4 point.

Let R be a perfect 𝔽-algebra equipped with the trivial norm. Let W(R) be the ring of p-typical Witt vectors over R equipped with the p-adic norm. At the level of nonarchimedean analytic spaces (in the sense of Berkovich), we demonstrate a close analogy between W(R) and the polynomial ring R[T] equipped with the Gauss norm, in which the role of the structure morphism from R to R[T] is played by the Teichmüller map. For instance, we show that the analytic space associated to R is a strong deformation retract of the space associated to W(R). We also show that each fiber forms a tree under the relation of pointwise comparison, and we classify the points of fibers in the manner of Berkovich’s classification of points of a nonarchimedean disk. Some results pertain to the study of p-adic representations of étale fundamental groups of nonarchimedean analytic spaces (i.e., relative p-adic Hodge theory).

For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function of A. This distribution is expected to correspond to taking characteristic polynomials of a uniform random matrix in some closed subgroup of USp(4); this Sato–Tate group may be obtained from the Galois action on any Tate module of A. We show that the Sato–Tate group is limited to a particular list of 55 groups up to conjugacy. We then classify A according to the Galois module structure on the ℝ-algebra generated by endomorphisms of (the Galois type), and establish a matching with the classification of Sato–Tate groups; this shows that there are at most 52 groups up to conjugacy which occur as Sato–Tate groups for suitable A and k, of which 34 can occur for k=ℚ. Finally, we present examples of Jacobians of hyperelliptic curves exhibiting each Galois type (over ℚ whenever possible), and observe numerical agreement with the expected Sato–Tate distribution by comparing moment statistics.

We complete our proof that given an overconvergent F-isocrystal on a variety over a field of positive characteristic, one can pull back along a suitable generically finite cover to obtain an isocrystal which extends, with logarithmic singularities and nilpotent residues, to some complete variety. We also establish an analogue for F-isocrystals overconvergent inside a partial compactification. By previous results, this reduces to solving a local problem in a neighborhood of a valuation of height 1 and residual transcendence degree zero. We do this by studying the variation of some numerical invariants attached to p-adic differential modules, analogous to the irregularity of a complex meromorphic connection. This allows for an induction on the transcendence defect of the valuation, i.e., the discrepancy between the dimension of the variety and the rational rank of the valuation.

Summary

We come now to the subject of metric properties of matrices over a field complete for a given norm. While this topic is central to our study of differential modules over nonarchimedean fields, it is based on ideas which have their origins largely outside number theory. We have thus opted to present the main points first in the archimedean setting and then to repeat the presentation for nonarchimedean fields.

The main theme is the relationship between the norms of the eigenvalues of a matrix, which are core invariants but depend on the entries of the matrix in a somewhat complicated fashion, and some less structured but more readily visible invariants. The latter are the singular values of a matrix, which play a key role in numerical linear algebra in controlling the numerical stability of certain matrix operations (including the extraction of eigenvalues). Their role in our work is similar.

Before proceeding, we set some basic notation and terminology for matrices.

Summary

It has been suggested several times in this book that the study of p-adic differential equations is deeply connected with the theory of p-adic cohomology for varieties over finite fields. In particular, the Frobenius structures arising on Picard–Fuchs modules, discussed in the previous chapter, appear within this theory.

In this chapter, we introduce a little of the theory of rigid p-adic cohomology, as developed by Berthelot and others. In particular, we illustrate the role played by the p-adic local monodromy theorem in a fundamental finiteness problem in the theory.

Isocrystals on the affine line

We start with a concrete description of p-adic cohomology in a very special case, namely the cohomology of “locally constant” coefficient systems on the affine line over a finite field. This is due to Crew [62].

Definition 23.1.1. Let k be a perfect (for simplicity) field of characteristic p > 0. Let K be a complete discrete (again for simplicity) nonarchimedean field of characteristic 0 with kK = k. An overconvergent F-isocrystal on the affine line over k (with coefficients in K) is a finite differential module with Frobenius structure on the ring A = A, ∪β>1K〈t/β〉, for some absolute Frobenius lift ϕ; as in Proposition 17.3.1 the resulting category is independent of the choice of Frobenius lift.

Definition 23.1.2. Let M be an overconvergent F-isocrystal on the affine line over k. Let ℛ be a copy of the Robba ring with series parameter t−1, so that we can identify A as a subring of ℛ.

Summary

In Part III we focus our attention specifically on p-adic ordinary differential equations (although most of our results apply also to complete nonarchimedean fields of residual characteristic 0). To do this with maximal generality, one would need first to introduce a category of geometric spaces over which to work. This would require a fair bit of discussion of either rigid analytic geometry, in the manner of Tate, or nonarchimedean analytic geometry in the manner of Berkovich, neither of which we want either to assume or introduce. Fortunately, since we only need to consider one-dimensional spaces, we can manage by working completely algebraically and considering differential modules over appropriate rings.

In this chapter, we introduce those rings and collect their basic algebraic properties. This includes the fact that they carry Newton polygons analogous to those for polynomials. Another key fact is that there is a form of the approximation lemma (Lemma 1.3.7) valid over some of these rings.

Notation 8.0.1. Throughout this part, let K be a field of characteristic 0 that is complete for a nontrivial nonarchimedean norm | · |. (The assumption of characteristic 0 is not used in this chapter; it will become crucial when we start discussing differential modules again.) Let p denote the characteristic of the residue field kK. We do not assume p > 0 (as the case p = 0 may be useful for some applications), but when p > 0 we do require the norm to be normalized in such a way that |p| = p−1.

Summary

In this chapter, we introduce Dwork's technique of Frobenius descent to analyze the generic radius of convergence and subsidiary radii of a differential module, in the range where Newton polygons do not apply. In one direction we introduce a somewhat refined form of the Frobenius antecedents introduced by Christol and Dwork. These fail to apply in an important boundary case; we remedy this by introducing the new notion of Frobenius descendants, which covers the boundary case.

Using these results, we are able to improve a number of results from Chapter 6 in the special case of differential modules over Fρ. For instance we get a full decomposition by spectral radius, extending the visible decomposition theorem (Theorem 6.6.1) and the refined visible decomposition theorem (Theorem 6.8.2). We will use these results again to study the variation of subsidiary radii, and decomposition by subsidiary radii, in the remainder of this part.

Notation 10.0.1. Throughout this chapter we retain Hypothesis 9.0.1. We also continue to use Fρ to denote the completion of K(t) for the ρ-Gauss norm viewed as a differential field with respect to d = d/dt, unless otherwise specified.

Summary

This book is an outgrowth of a course, taught by the author at MIT during fall 2007, on p-adic ordinary differential equations. The target audience was graduate students with some prior background in algebraic number theory, including exposure to p-adic numbers, but not necessarily with any background in p-adic analytic geometry (of either the Tate or Berkovich flavors).

Custom would dictate that ordinarily this preface would continue with an explanation of what p-adic differential equations are, and why they matter. Since we have included a whole chapter on this topic (Chapter 0), we will devote this preface instead to a discussion of the origin of the book, its general structure, and what makes it different from previous books on the subject.

The subject of p-adic differential equations has been treated in several previous books. Two that we used in preparing the MIT course, and to which we make frequent reference in the text, are those of Dwork, Gerotto, and Sullivan [80] and of Christol [42]. Another existing book is that of Dwork [78], but it is not a general treatise; rather, it focuses in detail on hypergeometric functions.

However, this book develops the theory of p-adic differential equations in a manner that differs significantly from most prior literature. Key differences include the following.

We limit our use of cyclic vectors. This requires an initial investment in the study of matrix inequalities (Chapter 4) and lattice approximation arguments (especially Lemma 8.6.1), but it pays off in significantly stronger results.

We introduce the notion of a Frobenius descendant (Chapter 10). This complements the older construction of Frobenius antecedents, particularly in dealing with certain boundary cases where the antecedent method does not apply.

Summary

In this chapter, we recall some basic facts about norms (absolute values), primarily of the nonarchimedean sort, on groups, rings, fields, and modules. We also briefly discuss the phenomenon of spherical completeness, which is peculiar to the nonarchimedean setting. Our discussion is not particularly comprehensive; the reader new to nonarchimedean analysis is directed to [191] for a fuller treatment.

Several proofs in this chapter make forward references to Chapter 2. There should be no difficulty in verifying the absence of circular references.

Convention 1.0.1. In this book, a ring means a commutative ring unless commutativity is suppressed explicitly by describing the ring as “not necessarily commutative” or implicitly by its usage in certain phrases, e.g., a ring of twisted polynomials (Definition 5.5.1).

Notation 1.0.2. For R a ring, we denote by R× the multiplicative group of units of R.

Norms on abelian groups

Let us start by recalling some basic definitions from analysis, before specializing to the nonarchimedean case.

Summary

In this chapter, we recall the traditional theory of Newton polygons for polynomials over a nonarchimedean field. In the process, we introduce a general framework which will allow us to consider Newton polygons in a wider range of circumstances; it is based on a version of Hensel's lemma that applies in not necessarily commutative rings. As a first application, we fill in a few missing proofs from Chapter 1.

Introduction to Newton polygons

We start with the possibly familiar notion of the Newton polygon associated with a polynomial over a nonarchimedean ring.

Definition 2.1.1. Let R be a ring equipped with a nonarchimedean submultiplicative (semi)norm | · |. For ρ > 0 and P = Σi Pi Ti ∈ R[T], define the width of P under the ρ-Gauss norm | · |ρ as the difference between the maximum and minimum values of i for which maxi{|Pi|ρi} is achieved.

Summary

In this chapter we construct a class of examples of differential modules on open annuli which carry Frobenius structures and hence are solvable at a boundary. These modules are derived from continuous linear representations of the absolute Galois group of a positive-characteristic local field.

We first construct a correspondence between Galois representations and differential modules over ℇ carrying a unit-root Frobenius structure. The basic mechanism for producing these modules is to tensor with a large ring carrying a Galois action and then take Galois invariants. This mechanism will reappear when we turn to p-adic Hodge theory, at which point we will attempt to simulate this situation using the Galois group of a mixed-characteristic local field. See Chapter 24.

Then we refine the construction to compare Galois representations having finite image of inertia with differential modules over ℇ± carrying a unit-root Frobenius structure; the main result here is an equivalence of categories due to Tsuzuki. It is generalized by the absolute case of the p-adic local monodromy theorem (Theorem 20.1.4 below) and indeed can be used together with the slope filtration theorem (Theorem 17.4.3) to prove the monodromy theorem in the absolute case. This result also has an analogue in p-adic Hodge theory; see Theorem 24.2.5.

We finally describe (without proof) a numerical relationship between the wild ramification of a Galois representation and the convergence of solutions of p-adic differential equations. Besides making explicit the analogy between the wild ramification of Galois representations and the irregularity of meromorphic differential systems, it also suggests an approach to higherdimensional ramification theory.